Pseudoinverse

Moore-Penrose Pseudoinverse

The pseudoinverse $A^+$ generalises the matrix inverse to non-square and singular matrices. It gives the minimum-norm least-squares solution to $Ax = b$:

$x = A^+ b$

For Full Column Rank ($m \geq n$, rank $n$)

When $A$ has full column rank, the pseudoinverse is:

$A^+ = (A^T A)^{-1} A^T$

This is exactly the least-squares formula.

For Square Invertible $A$

$A^+ = A^{-1}$

Example

$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad A^T A = \begin{pmatrix}1&0\\0&1\end{pmatrix}, \quad A^+ = (A^T A)^{-1} A^T = A^T = \begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}$

Your Task

Implement pseudoinverse(A) for matrices with full column rank using the formula $A^+ = (A^T A)^{-1} A^T$. The helper inv_square handles both 1×1 and 2×2 square matrices.